Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true if $|G:H|<32$ (see Corollary 4.33 of this paper). It's also true for the four first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here.